# Useful Facts :::{.definition title="Acyclic"} A chain complex $C$ is **acyclic** if and only if $H_*(C) = 0$. ::: :::{.proposition title="Algebra Facts"} \envlist - Free $\implies$ projective $\implies$ flat $\implies$ torsionfree (for finitely-generated \(R\dash\)modules) - Over $R$ a PID: free $\iff$ torsionfree ::: :::{.remark} Notational conventions: - Finite direct products: \( \bigoplus \) - Cohomological indexing: $C^i, \bd^i$ - Homological indexing: $C_i, \bd_i$ - Right-derived functors $R^iF$. - Come from left-exact functors - Require *injective* resolutions - Extend to the right: $0 \to F(A) \to F(B) \to F(C) \to L_1 F(A) \cdots$ - Left-derived functors $L_i F$. - Come from right-exact functors - Require *projective* resolutions - Extend to the left: $\cdots L_1F(C) \to F(A) \to F(B) \to F(C) \to 0$ - Colimits: - Examples: coproducts, direct limits, cokernels, initial objects, pushouts - Commute with left adjoints, i.e. $L(\colim F_i) = \colim LF_i$. - Examples of limits: - Products, inverse limits, kernels, terminal objects, pullbacks - Commute with right adjoints. i.e. $R(\colim F_i) = \colim RF_i$. ::: ## Hom and Ext :::{.proposition title="Basic properties of Hom"} \envlist - $\Hom_R(A, \wait)$ is: - Covariant - Left-exact - Is a functor that sends $f:X\to Y$ to $f_*: \Hom(A, X) \to \Hom(A, Y)$ given by $f_*(h) = f\circ h$. - Has right-derived functors $\Ext^i_R(A, B) \da R^i \Hom_R(A, \wait)(B)$ computed using *injective* resolutions. - $\Hom_R(\wait, B)$ is: - Contravariant - Right-exact - Is a functor that sends $f:X\to Y$ to $f^*: \Hom(Y, B) \to \Hom(X, B)$ given by $f^*(h) = h\circ f$. - Has left-derived functors $\Ext^i_R(A, B) \da L_i \Hom_R(\wait, B)(A)$ computed using *projective* resolutions. - For $N \in \bimod{R}{S'}$ and $M\in \bimod{R}{S}$, $\Hom_R(M, N) \in \bimod{S}{S'}$. - Mnemonic: the slots of $\Hom_R$ use up a left $R\dash$action. In the first slot, the right $S\dash$action on $M$ becomes a left $S\dash$action on Hom. In the second slot, the right $S'\dash$action on $N$ becomes a right $S'\dash$action on Hom. ::: :::{.proposition title="Basic Properties of Ext"} \envlist - $\Ext^{>1}(A, B) = 0$ for any $A$ projective or $B$ injective. ::: :::{.fact} A maps $A \mapsvia{f} B$ in $\rmod$ is injective if and only if $f(a) = 0_B \implies a = 0_A$. Monomorphisms are injective maps in $\rmod$. ::: :::{.proposition title="Recipe for computing $\Ext_R^i$"} Write $F(\wait) \da \Hom_R(A, \wait)$. This is left-exact and thus has right-derived functors $\Ext^i_R(A, B) \da R^iF(B)$. To compute this: - Take an *injective* resolution: \[ 1 \to B \mapsvia{\eps} I^0 \mapsvia{d^0} I^1 \mapsvia{d^1} \cdots .\] - Remove the augmentation $\eps$ and just keep the complex \[ I^\wait \da \qty{ 1 \mapsvia{d^{-1}} I^0 \mapsvia{d^0} I^1 \mapsvia{d^1} \cdots } .\] - Apply $F(\wait)$ to get a new (and usually **not exact**) complex \[ F(I)^\wait \da \qty{ 1 \mapsvia{\bd^{-1}} F(I^0) \mapsvia{\bd^0} F(I^1) \mapsvia{\bd^1} \cdots } ,\] where $\bd^i \da F(d^i)$. - Take homology, i.e. kernels mod images: \[ R^iF(B) \da { \ker d^i \over \im d^{i-1}} .\] Note that $R^0 F(B) \cong F(B)$ canonically: - This is defined as $\ker \bd^0 / \im \bd^{-1} = \ker \bd^0 / 1 = \ker \bd^0$. - Use the fact that $F(\wait)$ is left exact and apply it to the *augmented* complex to obtain \[ 1 \to F(B) \mapsvia{F(\eps)} F(I^0) \mapsvia{\bd^0} F(I^1) \mapsvia{\bd^1} \cdots .\] - By exactness, there is an isomorphism $\ker \bd^0 \cong F(B)$. ::: :::{.proposition title="Computing $\Hom_\ZZ(\ZZ, \ZZ/n)$"} $\phi: \Hom_{\ZZ}(\ZZ, \ZZ/n) \mapsvia{\sim} \ZZ/n$, where $\phi(g) \da g(1)$. - That this is an isomorphism follows from - Surjectivity: for each $\ell \in \ZZ/n$ define a map \[ \psi_y: \ZZ &\to \ZZ/n \\ 1 &\mapsto [\ell]_n .\] - Injectivity: if $g(1) = [0]_n$, then \[ g(x) = xg(1) = x[0]_n = [0]_n .\] - \(\ZZ\dash\)module morphism: \[ \phi(gf) \da \phi(g\circ f) \da (g\circ f)(1) = g(f(1)) = f(1)g(1) = \phi(g)\phi(f) ,\] where we've used the fact that $\ZZ/n$ is commutative. ::: :::{.proposition title="Common Hom Groups"} - $\Hom_\ZZ(\ZZ/m, \ZZ) = 0$. - $\Hom_\ZZ(\ZZ/m, \ZZ/n) = \ZZ/d$. - $\Hom_\ZZ(\QQ, \QQ) = \QQ$. ::: :::{.proposition title="Common Ext Groups"} - $\Ext_\ZZ(\ZZ/m, G) \cong G/mG$ - Use $1 \to \ZZ \mapsvia{\times m} \ZZ \mapsvia{} \ZZ/m \to 1$ and apply $\Hom_\ZZ(\wait, \ZZ)$. - $\Ext_\ZZ(\ZZ/m, \ZZ/n) = \ZZ/d$. ::: :::{.slogan} \envlist - In $\Ab$, direct colimits commute with finite limits. Inverse limits do not generally commute with finite colimits. - Left adjoints are right-exact with left-derived functors. Right adjoints are left-exact with right-derived functors. - Left adjoints commute with colimits: $L( \colim F) = \colim (L\circ F)$ ::: :::{.proposition title="Characterizations of Splittings"} TFAE in \( \rmod \): - A SES $0\to A\to B \to C\to 0$ is split. - ? ::: ## Tensor and Tor :::{.proposition title="Basic Properties of the Tensor Product"} \envlist - $A\tensor_R \wait$ is: - Covariant - Right-exact - Left-exact - Has left-derived functors $\Ext^i_R(A, B) \da L_i \Hom_R(\wait, B)(A)$ computed using *projective* resolutions. - $\wait\tensor_R B$ is: - Covariant - Right-exact - Has left-derived functors $\Ext^i_R(A, B) \da L_i \Hom_R(\wait, B)(A)$ computed using *projective* resolutions. - Tensor commutes with colimits: $(\colim A_i)\tensor_R M = \colim (A_i \tensor_R M)$. ::: :::{.proposition title="Basic Properties of Tor"} \envlist - $\Tor_n^R(A, B) = 0$ for either $A$ or $B$ flat. ::: :::{.fact} The most useful SES for proofs here: \[ 0 \to \ZZ \mapsvia{n} \ZZ \mapsvia{\pi} \ZZ/n \to 0 .\] ::: :::{.proposition title="Common Tensor Products"} \envlist - $\ZZ/n \tensor_\ZZ G \cong G/nG$ - $\ZZ/n \tensor_\ZZ \ZZ/m \cong \ZZ/d$. - $\QQ\tensor_\ZZ \ZZ/n \cong 0$. ::: :::{.proposition title="Common Tor Groups"} - $\Tor^\ZZ_1(\ZZ/n, G) \cong \ts{ h\in H \st nh = e }$ - $\Tor^\ZZ_1(\ZZ/n, \QQ) \cong 0$. - $\Tor^\ZZ_1(\ZZ/n, \ZZ/m) \cong \ZZ/d$. ::: ## Universal Properties :::{.proposition title="Universal Property of the Quotient for Groups"} If $f: G\to K$ and $H\normal G$ (so that $G/H$ is defined), then the map $f$ descends to the quotient if and only if $H \subseteq \ker(f)$. ::: :::{.proposition title="Kernels as pullbacks and cokernels as pushouts"} The kernel $\ker f$ of a morphism $f$ can be characterized as a cartesian square, and the cokernel $\coker f$ as a cocartesian square: \begin{tikzcd} K \\ & {\ker f} && \textcolor{rgb,255:red,92;green,92;blue,214}{A} && 0 \\ \\ & 0 && \textcolor{rgb,255:red,92;green,92;blue,214}{B} && {\coker f} \\ &&&&&& C \arrow[dotted, from=2-6, to=4-6] \arrow[from=2-4, to=2-6] \arrow["f"', color={rgb,255:red,92;green,92;blue,214}, from=2-4, to=4-4] \arrow["0"', dotted, from=4-4, to=4-6] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-6, to=2-4] \arrow["{\exists !}"', dashed, from=4-6, to=5-7] \arrow[curve={height=12pt}, from=4-4, to=5-7] \arrow[curve={height=-12pt}, from=2-6, to=5-7] \arrow[dotted, from=2-2, to=2-4] \arrow[from=4-2, to=4-4] \arrow[dotted, from=2-2, to=4-2] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=4-4] \arrow[curve={height=-12pt}, from=1-1, to=2-4] \arrow[curve={height=12pt}, from=1-1, to=4-2] \arrow["{\exists !}"', dashed, from=1-1, to=2-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMywxLCJBIixbMjQwLDYwLDYwLDFdXSxbNSwxLCIwIl0sWzMsMywiQiIsWzI0MCw2MCw2MCwxXV0sWzUsMywiXFxjb2tlciBmIl0sWzYsNCwiQyJdLFsxLDEsIlxca2VyIGYiXSxbMSwzLCIwIl0sWzAsMCwiSyJdLFsxLDMsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRvdHRlZCJ9fX1dLFswLDFdLFswLDIsImYiLDIseyJjb2xvdXIiOlsyNDAsNjAsNjBdfSxbMjQwLDYwLDYwLDFdXSxbMiwzLCIwIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzMsMCwiIiwwLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzMsNCwiXFxleGlzdHMgISIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDQsIiIsMCx7ImN1cnZlIjoyfV0sWzEsNCwiIiwwLHsiY3VydmUiOi0yfV0sWzUsMCwiIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzYsMl0sWzUsNiwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzUsMiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzcsMCwiIiwxLHsiY3VydmUiOi0yfV0sWzcsNiwiIiwxLHsiY3VydmUiOjJ9XSxbNyw1LCJcXGV4aXN0cyAhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) ::: ## Adjunctions :::{.definition title="Adjoints"} \todo[inline]{todo} ::: :::{.proposition title="Tensor-Hom Adjunction"} For a fixed $M\in \bimod{R}{S}$, there is an adjunction \[ \adjunction{ \wait \tensor_R M }{\Hom_S(M, \wait)}{ \modsright{R} } { \modsright{S} } ,\] so for $Y \in \bimod{A}{R}$ and $Z \in \bimod{B}{S}$, there is a (natural) isomorphism in \( \bimod{B}{A} \): \[ \Hom_S(X \tensor_R M, Z) \mapsvia{\sim} \Hom_R( X, \Hom_S(M, Z) ) .\] ::: :::{.proposition title="Forgetful Adjunctions"} Let \( F: \mods{R} \to \mods{\ZZ} \) be the forgetful functor, then there are adjunctions \[ \adjunction{F}{ \Hom_\ZZ(R, \wait)} {\mods{R} } {\mods{\ZZ} } \\ \\ \adjunction{R\tensor_\ZZ \wait }{F}{ \mods{\ZZ} }{ \mods{R} } .\] :::